On characterized subgroups of compact abelian groups

Abstract

Let X be a compact abelian group. A subgroup H of X is called characterized if there exists a sequence u=(un) of characters of X such that H=su(X), where su(X):={x∈X:(un,x)→0 in T}. Every characterized subgroup is an Fσδ-subgroup of X. We show that every Gδ-subgroup of X is characterized. On the other hand, X has non-characterized Fσ-subgroups.A subgroup H of X is said to be countable modulo compact (CMC) if H has a subgroup K such that it is a compact Gδ-subgroup of X and H/K is countable. It is proved that every characterized subgroup H of X is CMC if and only if X has finite exponent. This result gives a complete description of the characterized subgroups of compact abelian groups of finite exponent.For every sequence u=(un) of characters of X we define a refinement Xu of X, that is a Čech complete locally quasi-convex (almost metrizable) group. With the sequence u we associate the closed subgroup Hu of Xu and the natural projection πX:Xu→X such that πX(Hu)=su(X). This provides a description of the characterized subgroups of arbitrary compact abelian groups, extending the previously existing result from [25]. This description is new even in the case of metrizable compact groups.